There Exists No Globally Uniformly Convergent Reconstruction for the Paley–Wiener Space ${{cal PW}}_{\pi}^{1}$ of Bandlimited Functions Sampled at Nyquist Rate

The bandlimited interpolation of signals and the convergence behavior of the Shannon sampling series are discussed in order to show that it is desirable to have a uniformly convergent reconstruction, for as large a space of signals as possible. In this paper general sampling series are analyzed for the frequently utilized Paley-Wiener space PW _pi ¹, which is the largest space in the scale of Paley-Wiener spaces. The analysis is done not only for the Shannon sampling series, but for a whole class of axiomatically defined reconstruction processes. It is shown that for this very general class, which contains all common sampling series including the Shannon sampling series, a uniformly convergent reconstruction is not possible for the space PW _pi ¹. Moreover, a universal signal is identified that causes the divergence behavior for all sampling series. Finally, a lower and an upper bound are derived and used to describe the asymptotic behavior of the peak value of the finite sampling series.